It's a little difficult to see the differences really between MGTOW and "pick-up artists". They seem pretty similar to me, but look like they hate each other.
MGTOW is about being a better, stronger, more independent man in your own terms and explicitly ignoring feedback from woman as input as to what "better" means. For some guys, if you encounter women that happen to want to have sex with you, then go for. Others choose to remove the distraction entirely and ignore engaging in sex so they can focus on themselves.
Red Pill/PUA participants will sometimes look down at MGTOW because MGTOW has a lot of guys that aren't really dedicated to improving themselves. MGTOW guys can sometimes be bitter about their lack of sex life and try to use MGTOW to rationalize it and feel better about themselves in a sour-grapes sort of way.
If there's any "hate" on either side, they're taking it too seriously and not representative of the rational participants.
I don't think this is a fair characterization for what a lot of people refer to as "the PUA community". Accurate words would be confidence, independence, and leadership. Some people choose to view these attitudes as controlling/aggressive because when comparing these actions to themselves, it erodes their own confidence or self esteem. But that's their problem and they're trying to blame someone else for it.
The community certainly helped (and helps) a lot of men who suffer from confidence issues, but it also did/does so in a very flawed way, and threw everyone else (namely, all women) under the bus.